3.290 \(\int \frac{\sin ^{\frac{5}{2}}(x)}{\sqrt{\cos (x)}} \, dx\)
Optimal. Leaf size=143 \[ -\frac{1}{2} \sin ^{\frac{3}{2}}(x) \sqrt{\cos (x)}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right )}{4 \sqrt{2}}+\frac{3 \log \left (\tan (x)-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right )}{8 \sqrt{2}}-\frac{3 \log \left (\tan (x)+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right )}{8 \sqrt{2}} \]
[Out]
(-3*ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]])/(4*Sqrt[2]) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[C
os[x]]])/(4*Sqrt[2]) + (3*Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (3*Log[1 + (Sqr
t[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (Sqrt[Cos[x]]*Sin[x]^(3/2))/2
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Rubi [A] time = 0.113751, antiderivative size = 143, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used
= {2568, 2574, 297, 1162, 617, 204, 1165, 628} \[ -\frac{1}{2} \sin ^{\frac{3}{2}}(x) \sqrt{\cos (x)}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right )}{4 \sqrt{2}}+\frac{3 \log \left (\tan (x)-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right )}{8 \sqrt{2}}-\frac{3 \log \left (\tan (x)+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
Int[Sin[x]^(5/2)/Sqrt[Cos[x]],x]
[Out]
(-3*ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]])/(4*Sqrt[2]) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[C
os[x]]])/(4*Sqrt[2]) + (3*Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (3*Log[1 + (Sqr
t[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (Sqrt[Cos[x]]*Sin[x]^(3/2))/2
Rule 2568
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]
Rule 2574
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[(k*a*b)/f, Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos
[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Rule 297
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))
Rule 1162
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 1165
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rubi steps
\begin{align*} \int \frac{\sin ^{\frac{5}{2}}(x)}{\sqrt{\cos (x)}} \, dx &=-\frac{1}{2} \sqrt{\cos (x)} \sin ^{\frac{3}{2}}(x)+\frac{3}{4} \int \frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}} \, dx\\ &=-\frac{1}{2} \sqrt{\cos (x)} \sin ^{\frac{3}{2}}(x)+\frac{3}{2} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )\\ &=-\frac{1}{2} \sqrt{\cos (x)} \sin ^{\frac{3}{2}}(x)-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )\\ &=-\frac{1}{2} \sqrt{\cos (x)} \sin ^{\frac{3}{2}}(x)+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{8 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{8 \sqrt{2}}\\ &=\frac{3 \log \left (1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+\tan (x)\right )}{8 \sqrt{2}}-\frac{3 \log \left (1+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+\tan (x)\right )}{8 \sqrt{2}}-\frac{1}{2} \sqrt{\cos (x)} \sin ^{\frac{3}{2}}(x)+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{4 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{4 \sqrt{2}}\\ &=-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right )}{4 \sqrt{2}}+\frac{3 \log \left (1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+\tan (x)\right )}{8 \sqrt{2}}-\frac{3 \log \left (1+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+\tan (x)\right )}{8 \sqrt{2}}-\frac{1}{2} \sqrt{\cos (x)} \sin ^{\frac{3}{2}}(x)\\ \end{align*}
Mathematica [C] time = 0.0110258, size = 38, normalized size = 0.27 \[ \frac{2 \sin ^{\frac{7}{2}}(x) \cos ^2(x)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{7}{4};\frac{11}{4};\sin ^2(x)\right )}{7 \cos ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[x]^(5/2)/Sqrt[Cos[x]],x]
[Out]
(2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 7/4, 11/4, Sin[x]^2]*Sin[x]^(7/2))/(7*Cos[x]^(3/2))
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Maple [C] time = 0.171, size = 2667, normalized size = 18.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(x)^(5/2)/cos(x)^(1/2),x)
[Out]
-1/32*2^(1/2)*sin(x)^(3/2)*(12*cos(x)*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^
(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+4*2^(1/2
)-3*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi
((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+si
n(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(
1/2))-16*cos(x)^3*2^(1/2)-4*sin(x)^2*2^(1/2)+4*cos(x)^4*2^(1/2)+24*cos(x)^2*2^(1/2)-16*cos(x)*2^(1/2)+8*cos(x)
*sin(x)^2*2^(1/2)-4*cos(x)^2*sin(x)^2*2^(1/2)+12*cos(x)*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x
)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2
*2^(1/2))+3*I*sin(x)^4*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x
))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*I*sin(x)^4*(-(-1+cos(x)-sin(x)
)/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/s
in(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*cos(x)^4*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x)
)^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*I*co
s(x)^4*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*Ellipti
cPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+6*I*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*
((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2
-1/2*I,1/2*2^(1/2))-6*I*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos
(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-12*I*cos(x)^3*(-(-1+co
s(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x
)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+12*I*cos(x)^3*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+si
n(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(
1/2))+18*I*cos(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^
(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-18*I*cos(x)^2*(-(-1+cos(x)-sin(x))/
sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin
(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-12*I*cos(x)*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(
1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+12*I*cos(
x)*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(
(-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-6*cos(x)^2*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/
2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),
1/2-1/2*I,1/2*2^(1/2))-6*cos(x)^2*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2
)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+6*I*cos(x)^2
*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*Elli
pticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-6*I*cos(x)^2*sin(x)^2*(-(-1+cos(x)-sin(x))/si
n(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x
))^(1/2),1/2+1/2*I,1/2*2^(1/2))-12*I*cos(x)*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/si
n(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+12
*I*cos(x)*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(
1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-3*sin(x)^4*(-(-1+cos(x)-sin(x))/sin(
x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))
^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*sin(x)^4*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*
((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-3*cos(x)^4*(-(
-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+
cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*cos(x)^4*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+
sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2
^(1/2))-6*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(
1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-6*sin(x)^2*(-(-1+cos(x)-sin(x))/sin(
x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))
^(1/2),1/2+1/2*I,1/2*2^(1/2))+12*cos(x)^3*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)
*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+12*cos(x)^3*(
-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-
1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-18*cos(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(
x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/
2*2^(1/2))-18*cos(x)^2*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x
))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+12*cos(x)*(-(-1+cos(x)-sin(x))/s
in(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(
x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+12*cos(x)*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2
)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*(-(-1+co
s(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x
)-sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*I*(-(-1+cos(x)-sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(
x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((-(-1+cos(x)-sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2)))/(-1
+cos(x))^3/cos(x)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )^{\frac{5}{2}}}{\sqrt{\cos \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)^(5/2)/cos(x)^(1/2),x, algorithm="maxima")
[Out]
integrate(sin(x)^(5/2)/sqrt(cos(x)), x)
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Fricas [B] time = 5.43189, size = 1666, normalized size = 11.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)^(5/2)/cos(x)^(1/2),x, algorithm="fricas")
[Out]
-1/2*sqrt(cos(x))*sin(x)^(3/2) + 3/16*sqrt(2)*arctan(1/2*(2*cos(x)^3 - 2*cos(x)^2*sin(x) + sqrt(2)*sqrt(2*(sqr
t(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)*sqrt(cos(x))*sqrt(sin(x)) - sqr
t(2)*sqrt(cos(x))*sqrt(sin(x)) - 2*cos(x))/(cos(x)^3 + cos(x)^2*sin(x) - cos(x))) + 3/16*sqrt(2)*arctan(-1/2*(
2*cos(x)^3 - 2*cos(x)^2*sin(x) - sqrt(2)*sqrt(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) +
4*cos(x)*sin(x) + 1)*sqrt(cos(x))*sqrt(sin(x)) + sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) - 2*cos(x))/(cos(x)^3 + co
s(x)^2*sin(x) - cos(x))) - 3/16*sqrt(2)*arctan(-(sqrt(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(s
in(x)) + 4*cos(x)*sin(x) + 1)*(sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) + cos(x) + sin(x)) + sqrt(2)*sqrt(cos(x))*sqr
t(sin(x)))/(cos(x) - sin(x))) - 3/16*sqrt(2)*arctan(-(sqrt(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sq
rt(sin(x)) + 4*cos(x)*sin(x) + 1)*(sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) - cos(x) - sin(x)) + sqrt(2)*sqrt(cos(x))
*sqrt(sin(x)))/(cos(x) - sin(x))) - 3/32*sqrt(2)*log(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin
(x)) + 4*cos(x)*sin(x) + 1) + 3/32*sqrt(2)*log(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x))
+ 4*cos(x)*sin(x) + 1)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)**(5/2)/cos(x)**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )^{\frac{5}{2}}}{\sqrt{\cos \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)^(5/2)/cos(x)^(1/2),x, algorithm="giac")
[Out]
integrate(sin(x)^(5/2)/sqrt(cos(x)), x)